I actually didn't find any accuracy issues. Some quick minor suggestions:
[Professor]: 無理しないでいい 君は優秀な生徒だ
Don't overdo it. You're an excellent student, and we can't have you collapsing from working too hard.
I'm not at all familiar with this series, but a quick search on wikipedia reveals the "proper" romanization of エジー・トゥルス is Edge Turus. Again, I don't know anything about the series, but that's what it says. I think they might've decided on this spelling because "Tours" is commonly written as ツアーズ in Japanese.
As for the hard stuff from 324~326, the accuracy is fine. It's perfectly acceptable as is, but the flow isn't very good. I think every line that the professor has can be reworked. For example:
Just a very tiny bit early, the command to "dodge" or "catch" was sent from the brain, immediately before the ball was thrown.
So commands like "dodge" and "catch" were sent to the brain just a tiny bit before the ball was actually thrown.
The sentence structure of the original Japanese doesn't need to be preserved, as long as the meaning is still there. The structure isn't applicable to English anyways. I personally like to use italics for emphasis in long, annoying lines like this one, but that's just me.